Arcgis Geostatistical Analyst Tutorial Copyright © 2001, 2003–2006 ESRI All Rights Reserved

ArcGIS® 9 ArcGIS Geostatistical Analyst Tutorial Copyright © 2001, 2003–2006 ESRI All Rights Reserved. Printed in the United States of America. The information contained in this document is the exclusive property of ESRI. This work is protected under United States copyright law and the copyright laws of the given countries of origin and applicable international laws, treaties, and/or conventions. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage or retrieval system, except as expressly permitted in writing by ESRI. All requests should be sent to Attention: Contracts Manager, ESRI, 380 New York Street, Redlands, CA 92373-8100, USA.

The information contained in this document is subject to change without notice.

DATA CREDITS Carpathian Mountains data supplied by USDA Forest Service, Riverside, California, and is used here with permission. Radioceasium data supplied by International Sakharov Environmental University, Minsk, Belarus, and is used here with permission. Copyright © 1996. Air quality data for California supplied by California Environmental Protection Agency, Air Resource Board, and is used here with permission. Copyright © 1997. Radioceasium contamination in forest berries data supplied by the Institute of Radiation Safety “BELRAD”, Minsk, Belarus, and is used here with permission. Copyright © 1996.

CONTRIBUTING WRITERS Kevin Johnston, Jay M. Ver Hoef, Konstantin Krivoruchko, and Neil Lucas

DATA DISCLAIMER

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Attribution.pmd 1 05/23/2006, 9:39 AM ArcGIS Geostatistical Analyst Tutorial

IN THIS TUTORIAL With Geostatistical Analyst, you can easily create a continuous surface, or map, from measured sample points stored in a point-feature layer, raster • Exercise 1: Creating a surface layer, or by using polygon centroids. The sample points may be measurements using default parameters such as elevation, depth to the water table, or levels of pollution, as is the case • Exercise 2: Exploring your data in this tutorial. When used in conjunction with ArcMap, Geostatistical Analyst provides a comprehensive set of tools for creating surfaces that can be used • Exercise 3: Mapping ozone to visualize, analyze, and understand spatial phenomena. concentration Tutorial scenario • Exercise 4: Comparing models The U.S. Environmental Protection Agency is responsible for monitoring • Exercise 5: Mapping the atmospheric ozone concentration in California. Ozone concentration is mea- probability of ozone exceeding a sured at monitoring stations throughout the state. critical threshold The locations of the stations are shown here. The concentration levels of ozone are known for • Exercise 6: Producing the final map all of the stations, but we are also interested in knowing the level for every location in California. However, due to cost and practicality, monitoring stations cannot be everywhere. Geostatistical Analyst provides tools that make the best predictions possible by examining the relationships between all of the sample points and producing a continuous surface of ozone concentration, standard errors (uncertainty) of predictions, and probabilities that critical values are exceeded.

ch02_Tutorial.pmd 1 05/23/2006, 9:31 AM Introduction to the tutorial

The data you’ll need for this tutorial is included on the using the ESDA tools and working with the geostatistical Geostatistical Analyst installation disk. The datasets were parameters, you will be able to create a more accurate provided courtesy of the California Air Resources Board. surface. The datasets are: Many times it is not the actual values of some caustic Dataset Description health risk that is of concern, but rather if it is above some toxic level. If this is the case, immediate action must be ca_outline Outline map of California taken. The third surface you create will assess the probabil- ca_ozone_pts Ozone point samples (ppm) ity that a critical ozone threshold value has been exceeded. ca_cities Location of major California cities For this tutorial, the critical threshold will be if the maximum ca_hillshade A hillshade map of California average of ozone goes above 0.12 ppm in any eight-hour period during the year; then the location should be closely The ozone dataset (ca_ozone_pts) represents the 1996 monitored. You will use Geostatistical Analyst to predict the maximum eight-hour average concentration of ozone in probability of values complying with this standard. parts per million (ppm). (The measurements were taken daily and grouped into eight-hour blocks.) The original data This tutorial is divided into individual tasks that are designed has been modified for the purposes of the tutorial and to let you explore the capabilities of Geostatistical Analyst should not be taken to be accurate data. at your own pace. To get additional help, explore the ArcMap online Help system or sennnnnne Using ArcMap. From the ozone point samples (measurements), you will produce two continuous surfaces (maps), predicting the • Exercise 1 takes you through accessing Geostatistical values of ozone concentration for every location in the State Analyst and through the process of creating a surface of of California based on the sample points that you have. The ozone concentration to show you how easy it is to create first map that you create will simply use all default options a surface using the default parameters. to show you how easy it is to create a surface from your • Exercise 2 guides you through the process of exploring sample points. The second map that you produce will allow your data before you create the surface in order to spot you to incorporate more of the spatial relationships that are outliers in the data and to recognize trends. discovered among the points. When creating this second • Exercise 3 creates the second surface that considers map, you will use the ESDA tools to examine your data. more of the spatial relationships discovered in Exercise 2 You will also be introduced to some of the geostatistical and improves on the surface you created in Exercise 1. options that you can use to create a surface such as This exercise also introduces you to some of the basic removing trends and modeling spatial autocorrelation. By concepts of geostatistics.

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ch02_Tutorial.pmd 2 05/23/2006, 9:31 AM • Exercise 4 shows you how to compare the results of the two surfaces that you created in Exercises 1 and 3 in order to decide which provides the better predictions of the unknown values. • Exercise 5 takes you through the process of mapping the probability that ozone exceeds a critical threshold, thus creating the third surface. • Exercise 6 shows you how to present the surfaces you created in Exercises 3 and 5 for final display, using ArcMap functionality. You will need a few hours of focused time to complete the tutorial. However, you can also perform the exercises one at a time if you wish, saving your results after each exercise.

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ch02_Tutorial.pmd 3 05/23/2006, 9:31 AM Exercise 1: Creating a surface using default parameters

Before you begin you must first start ArcMap and enable 1 5 Geostatistical Analyst.

Starting ArcMap and enabling Geostatistical 7 Analyst 4 Click the Start button on the Windows taskbar, point to Programs, point to ArcGIS, and click ArcMap. In ArcMap, click Tools, click Extensions, and check Geostatistical Analyst. Click Close.

Adding the Geostatistical Analyst toolbar to ArcMap Click View, point to Toolbars, and click Geostatistical Analyst. Adding data layers to ArcMap 6 Once the data has been added, you can use ArcMap to display the data and, if necessary, to change the properties of each layer (symbology, and so on). The ca_outline layer is now displayed transparently with just the outline visible. This allows you to see the layers 1. Click the Add Data button on the Standard toolbar. that you will create in this tutorial underneath this layer. 2. Navigate to the folder where you installed the tutorial data (the default installation path is Saving your map C:\ArcGIS\ArcTutor\Geostatistics), hold down the Ctrl It is recommended that you save your map after each key, then click and highlight the ca_ozone_pts and exercise. ca_outline datasets. 7. Click the Save button on the Standard toolbar. 3. Click Add. You will need to provide a name for the map because 4. Click the ca_outline layer legend in the table of contents this is the first time you have saved it (we suggest to open the Symbol Selector dialog box. Ozone Prediction Map.mxd). To save in the future, click 5. Click the Fill Color dropdown arrow and click No Color. Save. 6. Click OK on the Symbol Selector dialog box.

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ch02_Tutorial.pmd 4 05/23/2006, 9:31 AM Creating a surface using the defaults Next you will create (interpolate) a surface of ozone 2 concentration using the default settings of Geostatistical 3 Analyst. You will use the ozone point dataset (ca_ozone_pts) as the input dataset and interpolate the ozone values at the locations where values are not known using ordinary kriging. You will click Next in many of the dialog boxes, thus accepting the defaults. Do not worry about the details of the dialog boxes in this exercise. Each dialog box will be revisited in later exercises. The intent of this exercise is to create a surface using the default options. 1. Click the Geostatistical Analyst toolbar, then click Geostatistical Wizard. 4 5

6. Click Next on the Geostatistical Method Selection dialog 1 box.

2. Click the Input Data dropdown arrow and click ca_ozone_pts. 3. Click the Attribute dropdown arrow and click the OZONE attribute. 4. Click Kriging in the Methods dialog box. 5. Click Next. By default, Ordinary Kriging and Prediction Map will be selected in the Geostatistical Method Selection dialog box. Note that having selected the method to map the ozone surface, you could click Finish here to create a surface 6 using the default parameters. However, steps 6 to 10 will expose you to many of the different dialog boxes.

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ch02_Tutorial.pmd 5 05/23/2006, 9:31 AM 7 8

The Semivariogram/Covariance Modeling dialog box allows The crosshairs show a location that has no measured value. you to examine spatial relationships between measured To predict a value at the crosshairs you can use the values points. You assume things that are close are more alike. at the measured locations. You know that the values of the The semivariogram allows you to explore this assumption. closer measured locations are more like the value of the The process of fitting a semivariogram model while captur- unmeasured location that you are trying to predict. The red ing the spatial relationships is known as variography. points in the above image are going to be weighted (or 7. Click Next. influence the unknown value) more than the green points since they are closer to the location you are predicting. Using the surrounding points, with the model fitted in the Semivariogram Modeling dialog box, you can predict a more accurate value for the unmeasured location. 8. Click Next.

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ch02_Tutorial.pmd 6 05/23/2006, 9:31 AM Q

9 W

The Cross Validation dialog box gives you some idea of “how well” the model predicts the values at the unknown locations. You will learn how to use the graph and understand the statistics in Exercise 4. 9. Click Finish. The Output Layer Information dialog box summarizes information on the method (and its associated parameters) that will be used to create the output surface. 10. Click OK. The predicted ozone map will appear as the top layer in This name change will help you distinguish this layer the table of contents. from the one you will create in Exercise 4. 11. Click the layer in the table of contents to highlight it, 12. Click save on the ArcMap Standard toolbar. then click again, and change the layer name to Notice that the interpolation continues into the ocean. You “Default”. will learn in Exercise 6 how to restrict the prediction surface to stay within California.

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ch02_Tutorial.pmd 7 05/23/2006, 9:31 AM Surface-fitting methodology You have now created a map of ozone concentration and completed Exercise 1 of the tutorial. While it is a simple task to create a map (surface) using Geostatistical Analyst, it is important to follow a structured process as shown below:

Exercise 1 Represent Add layers and display them in the ArcMap data view. the data Exercise 2 Investigate the statistical properties of your dataset. These tools can be used Explore to investigate the data, whether or not the intention is to create a surface. the data Exercise 3 Select a model to create a surface. The exploratory data phase will help in Fit a the selection of an appropriate model. model Exercise 3 Perform Assess the output surface. This will help you understand “how well” diagnostics the model predicts the unknown values. Exercise 4 If more than one surface is produced, the results can be Compare compared and a decision made as to which provides the better the models predictions of unknown values.

You will follow this structured process in the following exercises of the tutorial. In addition, in Exercise 5, you will create a surface of those locations that exceed a specified threshold and, in Exercise 6, you will create a final presentation layout of the results of the analysis performed in the tutorial. Note that you have already performed the first step of this process, representing the data, in Exercise 1. In Exercise 2, you will explore the data.

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ch02_Tutorial.pmd 8 05/23/2006, 9:31 AM Exercise 2: Exploring your data

In this exercise you will explore your data. As the struc- enabling you to examine the univariate (one-variable) tured process on the previous page suggests, to make better distribution of the dataset for each attribute. Next, you will decisions when creating a surface you should first explore explore the distribution of ozone for the ca_ozone_pts layer. your dataset to gain a better understanding of it. When 1. Click ca_ozone_pts, move it to the top of the table of exploring your data you should look for obvious errors in the contents, then place ca_outline underneath input sample data that may drastically affect the output ca_ozone_pts. prediction surface, examine how the data is distributed, look for global trends, etc. Geostatistical Analyst provides many data-exploration tools. In this tutorial you will explore your data in three ways: • Examine the distribution of your data. • Identify the trends in your data, if any. • Understand the spatial autocorrelation and directional influences. 1 If you closed the map after Exercise 1, click the File menu and click Open. In the dialog box, click the Look in box dropdown arrow and navigate to the folder where you 2. Click the Geostatistical Analyst toolbar, point to Explore saved the map document (Ozone Prediction Map.mxd). Data, and click Histogram. Click Open.

Examining the distribution of your data 2

Histogram The interpolation methods that are used to generate a surface give the best results if the data is normally distributed (a bell-shaped curve). If your data is skewed (lop- sided) you may choose to transform the data to make it normal. Thus, it is important to understand the distribution of You may wish to resize the Histogram dialog box so you your data before creating a surface. The Histogram tool can also see the map, as the following diagram shows. plots frequency histograms for the attributes in the dataset,

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ch02_Tutorial.pmd 9 05/23/2006, 9:31 AM 3. Click the Layer dropdown arrow and click The histogram shown above indicates that the data is ca_ozone_pts. unimodal (one hump) and fairly symmetric. It appears to be 4. Click the Attribute dropdown arrow and click OZONE. close to a normal distribution. The right tail of the distribution indicates the presence of a relatively small number of 5 6 sample points with large ozone concentration values. 5. Click the histogram bar with ozone values ranging from 0.162 to 0.175 ppm. The sample points within this range are highlighted on the map. Note that these sample points are located within the Los Angeles region. 6. Click to close the dialog box.

Normal QQPlot The QQPlot is where you compare the distribution of the data to a standard normal distribution, providing yet another measure of the normality of the data. The closer the points are to creating a straight line, the closer the distribution is to being normally distributed. 1. Click the Geostatistical Analyst toolbar, point to Explore 34 Data, and click Normal QQPlot.

The distribution of the ozone attribute is depicted by a histogram with the range of values separated into 1 10 classes. The relative proportion (density) of data within each class is represented by the height of each bar. Generally, the important features of the distribution are its central value, its spread, and its symmetry. As a quick check, if the mean and the median are approximately the same value, you have one piece of evidence that the data may be normally distributed.

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ch02_Tutorial.pmd 10 05/23/2006, 9:31 AM 2. Click the Layer dropdown arrow and click If the data did not exhibit a normal distribution in either the ca_ozone_pts. Histogram or the Normal QQPlot, it may be necessary to 3. Click the Attribute dropdown arrow and click OZONE. transform the data to make it comform to a normal distribution before using certain kriging interpolation techniques. 4 4. Click to exit the dialog box.

Identifying global trends in your data If a trend exists in your data, it is the nonrandom (determin- istic) component of a surface that can be represented by some mathematical formula. For instance, a gently sloping hillside can be represented by a plane. A valley would be represented by a more complex formula (a second-order polynomial) that creates a “U” shape. This formula may produce the representation of the surface you desire. However, many times the formula is too smooth to accurately depict the surface because no hillside is a perfect plane nor is a valley a perfect “U” shape. If the trend surface does not adequately portray your surface for your particular need, you may want to remove it and continue with your analysis, modeling the residuals, which is what 2 3 remains after the trend is removed. When modeling the residuals, you will be analyzing the short-range variation in A General QQPlot is a graph on which the quantiles from the surface. This is the part that isn’t captured by the two distributions are plotted versus each other. For perfect plane or the perfect “U”. two identical distributions, the QQPlot will be a straight line. The Trend Analysis tool enables you to identify the pres- Therefore, it is possible to check the normality of the ozone ence/absence of trends in the input dataset. data by plotting the quantiles of that data versus the quantiles of a standard normal distribution. From the 1. Click the Geostatistical Analyst toolbar, point to Explore Normal QQPlot above you can see that the plot is very Data, and click Trend Analysis. close to a straight line. The main departure from this line 2. Click the Layer dropdown arrow and click occurs at high values of ozone concentration (which were ca_ozone_pts. highlighted in the histogram plot so they are highlighted here also).

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ch02_Tutorial.pmd 11 05/23/2006, 9:31 AM East–West North–South 4. Click the Rotate Projection scroll bar and scroll left until trend line trend line the rotation angle is 30o. This rotation enables you to see the shape of the east–west trend more clearly. You can see that the projection actually exhibits an upside-down “U” shape. Because the trend is “U” shaped, a second-order polynomial is a good choice to use for the global trend. Even though the trend is being exhibited on the east–west projection plane, because we East–West rotated the points 30o, the actual trend is northeast to axis southwest. The trend seen is possibly caused by the fact that the pollution is low at the coast, but moving inland there North–South axis are large human populations that taper off again at the mountains. You will remove these trends in Exercise 4. 5. Click to exit the dialog box.

“U” shape trend 2 3 5 3. Click the Attribute dropdown arrow and click OZONE. Each vertical stick in the trend analysis plot represents the location and value (height) of each data point. The points are projected onto the perpendicular planes, an east–west and a north–south plane. A best-fit line (a polynomial) is drawn through the projected points, which model trends in specific directions. If the line were flat, this would indicate that there would be no trend. However, if you look at the 4 light green line in the image above, you can see it starts out with low values and increases as it moves east until it levels out. This demonstrates that the data seems to exhibit a strong trend in the east–west direction and a weaker one in the north–south direction.

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ch02_Tutorial.pmd 12 05/23/2006, 9:31 AM Understanding spatial autocorrelation and The Semivariogram/Covariance Cloud allows you to directional influences examine the spatial autocorrelation between the measured sample points. In spatial autocorrelation, it is assumed that 1. Click the Geostatistical Analyst toolbar, point to Explore things that are close to one another are more alike. The Data, and click Semivariogram/Covariance Cloud. Semivariogram/Covariance Cloud lets you examine this relationship. To do so, a semivariogram value, which is the difference squared between the values of each pair of locations, is plotted on the y-axis relative to the distance separating each pair on the x-axis. Each red dot in the Semivariogram/Covariance Cloud 1 represents a pair of locations. Since closer locations should be more alike, in the semivariogram the close locations (far left on the x-axis) should have small semivariogram values (low on the y-axis). As the distance between the pairs of locations increases (move right on the x-axis), the 2. Click the Layer dropdown arrow and click semivariogram values should also increase (move up on the ca_ozone_pts. y-axis). However, a certain distance is reached where the 3. Click the Attribute dropdown arrow and click OZONE. cloud flattens out, indicating that the relationship between the pairs of locations beyond this distance is no longer correlated. Looking at the semivariogram, if it appears that some data locations that are close together (near zero on the x-axis) have a higher semivariogram value (high on the y-axis) than you would expect, you should investigate these pairs of locations to see if there is the possibility that the data is inaccurate. 4. Click and drag the Selection pointer over these points to highlight them. (Use the following diagram as a guide. It is not important to highlight the exact points the diagram displays.)

2 3

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ch02_Tutorial.pmd 13 05/23/2006, 9:31 AM 5. Check Show Search Direction. 6. Click and move the directional pointer to any angle. The direction the pointer is facing determines which pairs of 4 data locations are plotted on the semivariogram. For example, if the pointer is facing an east–west direction, only the pairs of data locations that are east or west of one another will be plotted on the semivariogram. This enables you to eliminate pairs you are not interested in and to explore the directional influences on the data. 7. Click and drag the Selection tool along the values with the highest semivariogram values to highlight them on the plot and in the map. (Use the following diagram as a guide. It is not important to highlight the exact points in The pairs of sample locations that are selected in the the diagram or to use the same search direction.) semivariogram are highlighted on the map, and lines link the locations, indicating the pairing. 7 8 There are many reasons why the data values differ more among sample locations between the Los Angeles area and other areas. One possibility is that there are more cars in the Los Angeles area than in other areas, which will invariably produce more pollution, contributing to a higher ozone buildup in the Los Angeles area. Besides global trends that were discussed in the previous section, there may also be directional influences affecting the data. The reasons for these directional influences may not be known, but they can be statistically quantified. These directional influences will affect the accuracy of the surface you create in the next exercise. However, once you know if one exists, Geostatistical Analyst provides tools to account for it in the surface-creation process. To explore for a directional influence in the semivariogram cloud, you use the Search Direction tools. 6 5

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ch02_Tutorial.pmd 14 05/23/2006, 9:31 AM You will notice that the majority of the linked locations In this exercise we learned: (representing pairs of points on the map), regardless of 1. The ozone data is close to a normal distribution. They distance, correspond to one of the sample points from the are unimodal and fairly symmetrical around the mean/ Los Angeles region. Taking more pairs of points, at any median line as seen in the histogram. distance, into consideration, shows that it is not just pairs of points from the Los Angeles region out to the coast that 2. The Normal QQPlot reaffirmed that the data is normally have high semivariogram values. Many of the pairs of data distributed since the points in the plot created a fairly locations from the Los Angeles region to other inland areas straight line, and transformation is not necessary. also have high semivariogram values. This is because the 3. Using the Trend Analysis tool you saw that the data values of ozone in the Los Angeles area are so much higher exhibited a trend and, once refined, identified that the than anywhere else in California. trend would be best fit by a second-order polynomial in 8. Click to exit the dialog box. the southeast to northwest direction (330 degrees). 9. Click Selection and click Clear Selected Features to 4. From the Semiovariogram/Covariance Cloud we found clear the highlighted points on the map. that the high values of ozone concentration in Los Angeles create high semivariance values with locations nearby as well as far away. 5. The semivariogram surface indicates there is a spatial autocorrelation in the data. Knowing that there are no outlier (or erroneous) sample 9 points in the dataset and that the distribution is close to normal, you can proceed with confidence to the surface interpolation. Also, you will be able to create a more accurate surface because you know that there is a trend in the data that you can adjust for in the interpolation.

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ch02_Tutorial.pmd 15 05/23/2006, 9:31 AM Exercise 3: Mapping ozone concentration

In Exercise 1, you used the default parameters to map ozone concentration. However, you did not take into 2 account the statistical properties of the sample data. For 3 example, from exploring the data in Exercise 2, it appeared that the data exhibited a trend. This can be incorporated into the interpolation process. In this exercise you will: • Improve on the map of ozone concentration created in Exercise 1. • Be introduced to some basic geostatistical concepts. Again you will use the ordinary kriging interpolation method and will incorporate the trend in your model to create better predictions. 4 5 1. Click the Geostatistical Analyst toolbar and click Geo- From the exploration of your data in Exercise 2, you statistical Wizard. discovered that there was a global trend in your data. After 2. Click the Input Data dropdown and click ca_ozone_pts. refinement with the Trend Analysis tool, you discovered that 3. Click the Attribute dropdown arrow and click the a second-order polynomial seemed reasonable and the trend OZONE attribute. was from the southeast to the northwest. This trend can be represented by a mathematical formula and removed from 4. Click Kriging in the Methods box. the data. Once the trend is removed, the statistical analysis 5. Click Next. will be performed on the residuals or the short-range By default, Ordinary Kriging and Prediction are se- variation component of the surface. The trend will auto- lected. matically be added back before the final surface is created so that the predictions will produce meaningful results. By removing the trend, the analysis that is to follow will not be influenced by the trend, and once it is added back a more accurate surface will be produced.

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ch02_Tutorial.pmd 16 05/23/2006, 9:31 AM Northwest to southeast global trend 6

Southwest to northeast global trend

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6. On the Geostatistical Method Selection dialog box, click Trends should only be removed if there is justification for the Order of Trend Removal dropdown arrow and click doing so. The southwest to northeast trend in air quality can Second. be attributed to an ozone buildup between the mountains A second-order polynomial will be fitted because a and the coast. The elevation and the prevailing wind U-shaped curve was detected in the southwest to direction are contributing factors to the relatively low values northeast direction in the Trend Analysis dialog box in in the mountains and at the coast. The high concentration of Exercise 2. humans also leads to high levels of pollution between the mountains and coast. The northwest to southeast trend 7. Click Next on the Geostatistical Method Selection dialog varies much more slowly due to the higher populations box. around Los Angeles and extending to lesser numbers in San By default, Geostatistical Analyst maps the global trend Francisco. Hence we can justifiably remove these trends. in the dataset. The surface indicates the most rapid 8. Click Next on the Detrending dialog box. change in the southwest to northeast direction and a more gradual change in the northwest–southeast direction (causing the ellipse shape).

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ch02_Tutorial.pmd 17 05/23/2006, 9:31 AM Semivariogram/Covariance modeling 9. Type a new Lag Size value of 12000. In the Semivariogram/Covariance Cloud in Exercise 2, you 10. Click in the input box and type 10 for the Number of explored the overall spatial autocorrelation of the measured Lags. points. To do so, you examined the semivariogram, which Reducing the lag size means that you are effectively showed the difference-squared of the values between each zooming in to model the details of the local variation be- pair of points at different distances. The goal of tween neighboring sample points. You will notice that with a Semivariance/Covariance modeling is to determine the best smaller lag size, the fitted semivariogram (the yellow line) fit for a model that will pass through the points in the rises sharply and then levels off. The range is the distance semivariogram (the yellow line in the diagram). where it levels off. This flattening out of the semivariogram The semivariogram is a function that relates semivariance indicates that there is little autocorrelation beyond the (or dissimilarity) of data points to the distance that sepa- range. rates them. Its graphical representation can be used to provide a picture of the spatial correlation of data points with their neighbors. The Semivariogram/Covariance Modeling dialog box allows you to model the spatial relationship in the dataset. By default, optimal parameters for a spherical semivariogram model are calculated. Geostatistical Analyst first determines a good lag size for grouping semivariogram values. The lag size is the size of a distance class into which pairs of locations are grouped in order to reduce the large number of possible combinations. This is called binning. As a result of the binning, notice that there are fewer points in this semivariogram than the one in Exercise 2. A good lag Q distance can also help reveal spatial correlations. The dialog box displays the semivariogram values as a surface and as a scatterplot related to distance. Then it fits a spherical 9 semivariogram model (best fit for all directions) and its associated parameter values, which are typically called the nugget, range, and partial sill. By removing the trend, the semivariogram will model the spatial autocorrelation among data points without having to Try to fit the semivariogram at small lags (distances). It is consider the trend in the data. The trend will be automati- possible to use different bin sizes and refit the default cally added back to the calculations before the final surface spherical model by changing the lag size and number of is produced. lags.

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ch02_Tutorial.pmd 18 05/23/2006, 9:31 AM Available semivariogram Color scale models Empirical semivariogram values Associated Semivariogram value parameter values Fitted semivariogram model

Semivariogram surface

The color scale, which represents the calculated semivari- semivariogram values represent dissimilarity. For our ogram value, provides a direct link between the empirical example, the semivariogram starts low at small distances semivariogram values on the graph and those on the (things close together are more similar) and increases as semivariogram surface. The value of each “cell” in the distance increases (things get more dissimilar farther apart). semivariogram surface is color coded, with lower values Notice from the semivariogram surface that dissimilarity blue and green and higher values orange and red. The increases more rapidly in the southwest to northeast average value for each cell of the semivariogram surface is direction than in the southeast to northwest direction. plotted on the semivariogram graph. The x-axis on the Earlier, you removed a coarse-scale trend. Now it appears semivariogram graph is the distance from the center of the that there are directional components to the autocorrelation cell to the center of the semivariogram surface. The at finer scales, so we will model that next.

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ch02_Tutorial.pmd 19 05/23/2006, 9:31 AM Directional semivariograms R A directional influence will affect the points of the semivariogram and the model that will be fit. In certain directions closer things may be more alike than in other directions. Directional influences are called anisotropy, and Geostatistical Analyst can account for them. Anisotropy can be caused by wind, runoff, a geological structure, or a wide variety of other processes. The directional influence can be statistically quantified and accounted for when making your map. W You can explore the dissimilarity in data points for a certain E direction with the Search Direction tool. This allows you to examine directional influences on the semivariogram chart. It does not affect the output surface. The following steps show you how to achieve this. 13. Check Anisotropy. 11. Check Show Search Direction. Note the reduction in the number of semivariogram values. Only those points in The blue ellipse on the semivariogram surface indicates the the direction of the search are displayed. range of the semivariogram in different directions. In this case the major axis lies approximately in the NNW–SSE 12. Click and hold the cursor on the center line in the direction. Search Direction. Move the direction of the search tool. As you change the direction of the search, note how the Anisotropy will now be incorporated into the model to adjust semivariogram changes. Only the semivariogram for the directional influence of autocorrelation in the output surface values within the direction of the search are surface. plotted on the semivariogram chart above. To actually account for the directional influences on the semivariogram model for the surface calculations, you must calculate the anisotropical semivariogram or covariance model.

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ch02_Tutorial.pmd 20 05/23/2006, 9:31 AM Range

T 15. Type the following parameters for the Search direction to make the directional pointer coincide with the major 14. Type the following parameters for the Search direction axis of the anisotropical ellipse: to make the directional pointer coincide with the minor Angle Direction: 340.0 axis of the anisotropical ellipse: Angle Tolerance: 45.0 Angle Direction: 236.0 Bandwidth (lags): 3.0 Angle Tolerance: 45.0 The semivariogram model increases more gradually, then Bandwidth (lags): 3.0 flattens out. The range in this direction is 114 km. Note that the shape of the semivariogram curve increases The plateau that the semivariogram models reach in both more rapidly to its sill value. The x- and y-coordinates are in steps 14 and 15 is the same and is known as the sill. The meters, so the range in this direction is approximately range is the distance at which the semivariogram model 74 km. reaches its limiting value (the sill). Beyond the range, the dissimilarity between points becomes constant with increased lag distance. The lag is defined by the distance

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ch02_Tutorial.pmd 21 05/23/2006, 9:31 AM Range

Sill

Nugget Lag distance

Anisotropical ellipse

between pairs of points. Points separated by a lag distance 16. Click Next. greater than the range are spatially uncorrelated. The Now you have a fitted model to describe the spatial auto- nugget represents measurement error and/or microscale correlation, taking into account detrending and directional variation (variation at spatial scales too fine to detect). It is influences in the data. This information, along with the possible to estimate the measurement error if you have configuration and measurements of locations around the multiple observations per location, or you can decompose prediction location, is used to make a prediction. But how the nugget into measurement error and microscale variation should man-measured locations be used for the calcula- by checking the Nugget Error Modeling check box. tions?

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ch02_Tutorial.pmd 22 05/23/2006, 9:32 AM Searching neighborhood ample, four locations (red) have weights of more than 10 percent. The larger the weight, the more impact that It is common practice to limit the data used by defining a location will have on the prediction of unknown values. circle (or ellipse) to enclose the points that are used to predict values at unmeasured locations. 17. Click inside the graph view to select a prediction location (where the crosshairs meet). Note the change Additionally, to avoid bias in a particular direction, the circle in the selection of data location (together with their (or ellipse) can be divided into sectors from which an equal associated weights) that will be used for calculating the number of points are selected. By using the Searching value at the prediction location. Neighborhood dialog box, you can specify the number of 18. For the purpose of this tutorial, type the following points (a maximum of 200), the radius (or major/minor axis), coordinates in the Test Location input boxes: and the number of sectors of the circle (or ellipse) to be used for prediction. X = -2044968 and Y = 208630.37. 19. Check the Shape check box and type 90 in the Angle The points highlighted in the data view window give an input box. Notice how the shape changes. However, to indication of the weights that will be associated with each account for the directional influences, change the angle location in the prediction of unknown values. In this ex- back to 338.1.

In each sector of the search neighborhood, the number of points used to predict a value at an unmeasured Locations used and location associated weights In each sector of the search neighborhood, the minimum Sector of search number of points to be used neighborhood Geometry and number of Crosshairs define the sectors used in the search location prediction

Perimeter of search O neighborhood

Preview surface or I neighbors S

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ch02_Tutorial.pmd 23 05/23/2006, 9:32 AM 20. Uncheck the Shape check box—Geostatistical Analyst will use the default values (calculated in the Semivariogram/Covariance dialog earlier). 21. Click Next on the Searching Neighborhood dialog box. Before you actually create the surface, you next use the Cross Validation dialog box to perform diagnostics on the parameters to determine “how good” your model will be.

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ch02_Tutorial.pmd 24 05/23/2006, 9:32 AM Cross-validation In addition to visualizing the scatter of points around this 1:1 line, a number of statistical measures can be used to Cross-validation gives you an idea of “how well” the model assess the model’s performance. The objective of cross- predicts the unknown values. validation is to help you make an informed decision about For all points, cross-validation sequentially omits a point, which model provides the most accurate predictions. For a predicts its value using the rest of the data, and then model that provides accurate predictions, the mean error compares the measured and predicted values. The calcu- should be close to 0, the root-mean-square error and lated statistics serve as diagnostics that indicate whether average standard error should be as small as possible (this the model is reasonable for map production. is useful when comparing models), and the root-mean- square stardardized error should be close to 1. Cross-validation Here the term “prediction error” is used for the difference Line of best fitscatter plot 1:1 Line between the prediction and the actual measured value. For a model that provides accurate predictions, the mean prediction error should be close to 0 if the predictions are unbiased, the root-mean-square standardized prediction error should be close to 1 if the standard errors are accurate, and the root-mean-square prediction error should be small if the predictions are close to the measured values. The Cross Validation dialog box also allows you to display scatterplots that show the Error, Standardized Error, and QQPlot for each data point.

Summary Results from statistics cross-validation exercise

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ch02_Tutorial.pmd 25 05/23/2006, 9:32 AM 22. Click the QQPlot tab to display the QQPlot. The Output Layer Information dialog box provides a From the QQPlot you can see that some values fall slightly summary of the model that will be used to create a above the line and some slightly below the line, but most surface. points fall very close to the straight dashed line, indicating that prediction errors are close to being normally distributed. 23. To highlight the location for a particular point, click on the row that relates to the point of interest in the table. The selected point is highlighted in green on the scattergram. 24. Optionally, click Save Cross Validation to save the table for further analysis of the results.

Selected point

26. Click OK. F The predicted ozone map will appear as the top layer in ArcMap.

G H

25. Click Finish.

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ch02_Tutorial.pmd 26 05/23/2006, 9:32 AM By default, the layer assumes the name of the kriging value ± 2 times the prediction standard error if data are method used to produce the surface (e.g., Ordinary Krig- normally distributed. Notice in the Prediction Standard Error ing). surface that locations near sample points generally have 27. Click the layer name to highlight it, then click it again lower error. and change it to “Trend removed”. The surface you created in Exercise 1 simply used the defaults of Geostatistical Analyst, with no consideration of Z K L trends in the surface, of using smaller lag sizes, or of using an anisotropic semivariogram model. The prediction surface you created in this exercise took into consideration the global trends in the data, adjusted the lag size, and adjusted for the local directional influence (anisotropy) in the semivariogram. In Exercise 4, you will compare the two models to see which one provides a better prediction of unknown values. Note: Once again, you see that the interpolation continues into the ocean. You will learn in Exercise 6 how to restrict the prediction surface to stay within California.

You can also create a Prediction Standard Error surface to examine the quality of the predictions. 28. Right-click on the “Trend removed” layer that you created and click on Create Prediction Standard Error Map. 29. Click Save on the Standard toolbar. The Prediction Standard Errors quantify the uncertainty for each location in the surface that you created. A simple rule of thumb is that 95 percent of the time, the true value of the surface will be within the interval formed by the predicted

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ch02_Tutorial.pmd 27 05/23/2006, 9:32 AM Exercise 4: Comparing models

Using Geostatistical Analyst, you can compare the results of two mapped surfaces. This allows you to make an informed decision as to which provides more accurate predictions of ozone concentration based on cross-validation statistics. 1. Right-click the “Trend removed” layer, point to Com- pare. You will be comparing the “Trend removed” layer with the Default layer you created in Exercise 2.

3. Right-click the Default layer and click Remove. 1

Because the root-mean-square prediction error is smaller for the Trend removed layer, the root-mean-square standardized prediction error is closer to one for the Trend removed layer, and the mean prediction error is also closer 4. Click the Trend removed layer and move it to the bottom to zero for the Trend removed layer, you can state with of the table of contents so that you can see the sample some evidence that the Trend removed model is better and points and outline of California. more valid. Thus, you can remove the default layer since 5. Click Save on the Standard toolbar. you no longer need it. You have now identified the best prediction surface, but 2. Click Close on the Cross Validation Comparison dialog there may be other types of surfaces that you might wish to box. create.

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ch02_Tutorial.pmd 28 05/23/2006, 9:32 AM Exercise 5: Mapping the probability of ozone exceeding a critical threshold

In Exercises 1 and 3 you used ordinary kriging to map 5. Click Next on the Choose Input Data and Method dialog ozone concentration in California using different param- box. eters. In the decision making process, care must be taken in 6. Click Indicator Kriging; notice that Probability Map is using a map of predicted ozone for identifying unsafe areas selected. because it is necessary to understand the uncertainty of the predictions. For example, suppose the critical threshold 7. Set the Primary Threshold Value to 0.12. ozone value is 0.12 ppm for an eight-hour period, and you 8. Click the Exceed radial button to select it. would like to decide if any locations exceed this value. To 9. Click Next on the Geostatistical Method Selection dialog aid the decision making process, you can use Geostatistical box. Analyst to map the probability that ozone values exceed the threshold. While Geostatistical Analyst provides a number of methods that can perform this task, for this exercise you will use the indicator kriging technique. This technique does not require the dataset to conform to a particular distribution. The data 6 values are transformed to a series of 0s and 1s according to whether the values of the data are below or above a 8 threshold. If a threshold above 0.12 ppm is used, any value below this threshold will be assigned a value of 0, whereas 7 the values above the threshold will be assigned a value of 1. Indicator kriging then uses a semivariogram model that is calculated from the transformed 0–1 dataset. 9 1. Click the Geostatistical Analyst toolbar and click Geo- statistical Wizard. 2. Click the Layer dropdown arrow and click 10. Click Next on the Additional Cutoffs Selection dialog ca_ozone_pts. box. 3. Click the Attribute dropdown arrow and click the 11. Click Anisotropy to account for the directional nature of OZONE attribute. the data. 4. Click Kriging in the method box. 12. Type 25000 for the lag size and 10 for the number of lags.

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ch02_Tutorial.pmd 29 05/23/2006, 9:32 AM 13. Click Next on the Semivariogram/Covariance Modeling The measured and indicator columns display the actual and dialog box. transformed values for each sample location. The indicator 14. Click Next on the Searching Neighborhood dialog box. prediction values can be interpreted as the probability of exeeding the threshold. The indicator prediction values are The blue line represents the threshold value (0.12 ppm). calculated using the semivariogram modeled from the Points to the left have an indicator-transform value of 0, binary (0,1) data, created as indicator transformations of whereas points to the right have an indicator-transform your original data. Cross-validation sequentially omits a value of 1. point and then calculates indicator prediction values for 15. Click and scroll right until the Measured, Indicator, and each. Indicator Prediction columns are displayed. For example, the highest measured value is 0.1736. If this 16. Click and highlight a row in the table with an indicator location had not actually been measured, a prediction of value of 0. That point will be highlighted in green on the about an 85 percent chance that it was above the threshold scattergraph, to the left of the blue threshold line. based on the indicator kriging model would have been made. 17. Click Finish on the Cross Validation dialog box. 18. Click OK on the Output Layer Information dialog box. The probability map will appear as the top layer in the ArcMap data view. The map displays the indicator prediction values, interpreted as the probability that the threshold value of 0.12 ppm was exceeded on one or more days in the U year 1996. Y

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ch02_Tutorial.pmd 30 05/23/2006, 9:32 AM P

It is clear from the map that near Los Angeles the probability of values exceeding our threshold (staying, on average, below 0.12 ppm for every eight-hour period during the year) is likely. 19. Click and hold the Indicator Kriging layer. Drag the layer and reposition between the ca_outline and trend removed layers. Click Save on the Standard toolbar to save your map. Exercise 6 will show you how you can use the functionality within ArcMap to produce a cartographically pleasing map of the prediction surface that you created in Exercise 3 and the probability surface that you created in this exercise.

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ch02_Tutorial.pmd 31 05/23/2006, 9:32 AM Exercise 6: Producing the final map

You will now produce a final map for presentation. You will 2 use ArcMap to produce a final output map in which the prediction and the probability surfaces will be displayed. 4 3 Displaying both surfaces You can change the display of the probability map so you will be able to see both the prediction and the probability maps at the same time. The probability levels will be displayed as a contour map. 1. Right-click the Indicator Kriging layer. Click Properties. 2. Click the Symbology tab. 5 3. Uncheck the Filled Contours check box, then check the Contours check box. 4. Click the Color Ramp dropdown arrow and choose an alternative color ramp. 5. Click OK. You now see both the probability map (the contours) and the prediction map as the diagram to the right shows.

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ch02_Tutorial.pmd 32 05/23/2006, 9:32 AM Extrapolating ozone values Clipping the layers to the California State outline By default, Geostatistical Analyst interpolates the value of You will now clip the layers to the ca_outline layer as you the selected variable at any location that lies within the area are only interested in mapping the ozone levels within the defined by the north–south and east–west limits of the State of California and this will produce a more appealing sample point data. However, the map of predicted ozone map. does not cover the geographical extent of California (the 1. Right-click Layers and click Properties. ca_outline layer). To overcome this problem you will extrapolate values (predict values outside the default bounding box) for both surfaces. 1. Right-click the Indicator Kriging layer in the table of contents and click Properties. Click the Extent tab. In Set the extent to box select a custom extent entered below and type the following values for the Visible Extent, then click OK: 1 Left: -2400000 Right: -1600000 Top: 860000 Bottom: -400000 Repeat this step for the Trend removed layer. 2. Click the Data Frame tab. 3. Check the Enable Clip to Shape check box. 4. Click Specify Shape. 5. Click Outline of Features. 6. Click the Layer dropdown and click ca_outline. 7. Click OK. 8. Click OK to close the Data Frame Properties dialog box.

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ch02_Tutorial.pmd 33 05/23/2006, 9:32 AM The clipped map should look like the following diagram. 2

3 4

8 Locating the City of Los Angeles 6 1. Click the Add Data button on the Standard toolbar. 2. Navigate to the folder where you installed the tutorial data (the default installation path is 5 C:\ArcGIS\ArcTutor\Geostatistics), then click ca_cities. 3. Click Add. A map of the location of cities in California will be displayed.

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ch02_Tutorial.pmd 34 05/23/2006, 9:32 AM 4. Right-click the ca_cities layer and click Open Attribute 7 Table. 5. Scroll through the table and find the AreaName called Los Angeles. Click this row. The City of Los Angeles is highlighted on the map. 6. Click to close the attribute table. 6

Create a layout 1. Click View on the Main menu and click Layout View. 2. Click the map to highlight it. 3. Click and drag the bottom-left corner of the Data Frame to resize the map.

7. Click the Zoom In tool on the Tools toolbar and zoom in on the City of Los Angeles. Notice that the area with the highest ozone concentration is actually located just to the east of Los Angeles. 3

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ch02_Tutorial.pmd 35 05/23/2006, 9:32 AM 4. Click Insert on the Main menu and click Data Frame. A new data frame is inserted on the map. You can now copy all the layers in the first data frame into this one in order to display a map of ozone values for the whole of California alongside the ozone map, which zooms in on the Los Angeles area.

5 6

Follow steps 5 and 6 for all the other layers. 7. Click and drag the New Data Frame to fit the whole page.

5. Right-click the Trend removed layer and click Copy. 6. Right-click the New Data Frame in the table of contents and click Paste Layer(s).

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ch02_Tutorial.pmd 36 05/23/2006, 9:32 AM 8. Click the Full Extent button on the Tools toolbar to view 4. Click ca_hillshade and move it to the bottom of the table the full extent of the map in the New Data Frame. of contents. 9. Right-click the New Data Frame and click Properties. 5. Right-click the Trend removed layer in the New Data 10. Click the Data Frame tab and, as you did for the first Frame table of contents and click Properties. Data Frame, check Enable Clip to Shape and click the Specify Shape button. Choose ca_outline as the layer to clip to, then click OK. 6 7 Adding a hillshade and transparency 1. Right-click the New Data Frame and click Add Data.

6. Click the Display tab. 7. Type 30 for the percentage of transparency. 8. Click OK. The hillshade should now partially display underneath the Trend removed layer.

2. Navigate to the folder where you installed the tutorial Adding map elements data (the default installation path is C:\ArcGIS\ArcTutor\Geostatistics), then click 1. Click Insert on the Main menu and click Legend. ca_hillshade. 2. Move the legend to the bottom-left corner of the layout. 3. Click Add. 3. Optionally, click Insert and add a North arrow, a Scale A hillshade map of California will be displayed. bar, and Text.

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ch02_Tutorial.pmd 37 05/23/2006, 9:32 AM The following diagram shows the type of finished map you The map shows that the area east of Los Angeles has the could produce using the functionality of ArcMap. Refer to highest predicted levels of ozone and the highest probability Using ArcMap if necessary to learn about inserting of exceeding the critical average threshold (0.12 ppm) in at elements into the layout. least one eight-hour period during 1996. Since this is the case in the analysis (but remember the original data has been altered), you may wish to focus on these areas and analyze time series measurements of ozone to accurately identify the areas at potential risk.

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ch02_Tutorial.pmd 38 05/23/2006, 9:32 AM